Manuel de León

Lagrangian submanifolds and dynamics on Lie algebroids

In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, we give a Hamiltonian description of mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits us to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange–Poincare (Hamilton–Poincare) equations are the Euler–Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange–Poincare (Hamilton–Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.

On the geometry of non‐holonomic Lagrangian systems

We present a geometric framework for non‐holonomic Lagrangian systems in terms of distributions on the configuration manifold. If the constrained system is regular, an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics. If the constrained system is singular, we develop a constraint algorithm which is very similar to that developed by Dirac and Bergmann, and later globalized by Gotay and Nester. Special attention to the case of constrained systems given by connections is paid. In particular, we extend the results of Koiller for Caplygin systems. An application to the so‐called non‐holonomic geometry is given.

Non-holonomic Lagrangian systems in jet manifolds

A geometrical setting in terms of jet manifolds is developed for time-dependent non-holonomic Lagrangian systems. An almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics. A constrained Poincare - Cartan 2-form is defined. If the non-holonomic system is singular, a constraint algorithm is constructed. Special attention is devoted to Caplygin systems and a reduction theorem is proved.

A SURVEY OF LAGRANGIAN MECHANICS AND CONTROL ON LIE ALGEBROIDS AND GROUPOIDS

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

Hamiltonian systems on k-cosymplectic manifolds

The Hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories. To do this, the notion of k-cosymplectic manifold is introduced, and a suitable Hamiltonian formalism is developed so that the field equations for scalar and vector Hamiltonian functions are derived.

Reduction of nonholonomic mechanical systems with symmetries

Abstract A geometric reduction procedure is presented for Lagrangian systems subjected to nonlinear nonholonomic constraints in the presence of symmetries. Our approach is based on a geometrical method which enables one to deduce the constrained dynamics from the unconstrained one by projection.

Geometrical theory of uniform cosserat media

A geometric description of generalized Cosserat media is presented in terms of non-holonomic frame bundles of second order. A non-holonomic Ḡ-structure is constructed by using the smooth uniformity of the material and its integrability is proved to be equivalent to the homogeneity of the body. If the material enjoys global uniformity, the theory of linear connections in frame bundles permits to express the inhomogeneity by means of some tensor fields.

A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories

We construct a geometrical formulation for first order classical field theories in terms of fibered manifolds and connections. Using this formulation, a constraint algorithm for singular field theories is developed. This algorithm extends the constraint algorithm in mechanics.

k-cosymplectic manifolds and Lagrangian field theories

A geometrical description of classical field theories of first order is given. The underlying k-cosymplectic structure permits to derive the corresponding field equations.

Towards a Hamilton–Jacobi theory for nonholonomic mechanical systems

In this paper, we obtain a Hamilton–Jacobi theory for nonholonomic mechanical systems. The results are applied to a large class of nonholonomic mechanical systems, the so-called Caplygin systems.